Weyl solutions and j-selfadjointness for Dirac operators

Abstract

We consider a non-selfadjoint Dirac-type differential expression(0.1)D(Q)y:=Jndydx+Q(x)y, with a non-selfadjoint potential matrix Q∈Lloc1(I,Cn×n) and a signature matrix Jn=−Jn−1=−Jn⁎∈Cn×n. Here I denotes either the line R or the half-line R+. With this differential expression one associates in L2(I,Cn) the (closed) maximal and minimal operators Dmax(Q) and Dmin(Q), respectively. One of our main results for the whole line case states that Dmax(Q)=Dmin(Q) in L2(R,Cn). Moreover, we show that if the minimal operator Dmin(Q) in L2(R,Cn) is j-symmetric with respect to an appropriate involution j, then it is j-selfadjoint. Similar results are valid in the case of the semiaxis R+. In particular, we show that if n=2p and the minimal operator Dmin+(Q) in L2(R+,C2p) is j-symmetric, then there exists a 2p×p-Weyl-type matrix solution Ψ(z,⋅)∈L2(R+,C2p×p) of the equation Dmax+(Q)Ψ(z,⋅)=zΨ(z,⋅). A similar result is valid for the expression (0.1) whenever there exists a proper extension A˜ with dim(domA˜/domDmin+(Q))=p and nonempty resolvent set. In particular, it holds if a potential matrix Q has a bounded imaginary part. This leads to the existence of a unique Weyl function for the expression (0.1). The main results are proven by means of a reduction to the self-adjoint case by using the technique of dual pairs of operators. The differential expression (0.1) is of significance as it appears in the Lax formulation of the vector-valued nonlinear Schrödinger equation.